Visitors to a website click on an add with fixed unknown frequency $p$. You collect a sample of $n$ visits and compute the clickrate $\hat{p}$. Find the minimum $n$ such that $$P(|\hat{p}-p|<\delta)\geq .95$$
My question is: is it possible to answer this without knowing $p$? I have that:
$\hat{p} \sim \frac{1}{n}X$
where $X\sim\text{Bin}(n,p)$
Using the normal approximation we can say $X\approx Y$ where $Y\sim N(np,np(1-p))$, and so we can say $\hat{p}$ is approximately distributed as $N\left(p, \left(\frac{\sqrt{p(1-p)}}{\sqrt{n}}\right)^2\right)$. Then
$\begin{align}P(|\hat{p}-p|<\delta) &= 2P(\hat{p}-p<\delta) \\ &=2P\left(\frac{\sqrt{n}(\hat{p}-p)}{\sqrt{p(1-p)}} < \delta\frac{\sqrt{n}}{\sqrt{p(1-p)}}\right) \\ &= 2\ \Phi\left(\delta\frac{\sqrt{n}}{\sqrt{p(1-p)}}\right) \end{align}$
Setting the last line equal to $.95$ you can solve for $n$, but the answer is in terms of $p$ which is unknown. Is there a way to get rid of $p$?